Polyhedra with Equal Faces and Equal Vertex Figures

Interestingly, the nine regular solids
are not the only polyhedra with equal faces and equal vertex
figures. They would be if we required the faces to be regular polygons,
but if we allow irregular polygons, a number of other examples are known.

If you "stretch" a regular tetrahedron by pulling two opposite
edges apart while keeping them orthogonal, the result is a kind of tetragonal
disphenoid. It has four congruent isosceles faces. If you "squish"
rather than stretch, you get one like this.

If you also rotate the two opposite edges relative to each other about
their common 2-fold axis, the result is a rhombic
disphenoid. It has four congruent scalene faces. At each vertex, one
of each type of face angle meets.

In both cases, the disphenoid has a net which is a triangle with edge
lengths twice those of a face, divided (at the three edge midpoints) into
four congruent triangles (the central one being "upside down"
relative to the other three).